Comportement asymptotique des marches aleatoires associees aux polynomes de Gegenbauer et applications

Random walks on N associated with orthogonal polynomials have properties similar to classical random walks on . In fact such processes have independent increments with respect to a hypergroup structure on with a convolution and a Fourier transform which is the basic tool for their study. We illustrate these ideas by giving a description of the asymptotic behaviour (CLT and ILL) of the random walks associated with Gegenbauer's polynomials. Moreover we can then use these random walks as a reference scale to deduce asymptotic properties of other Markov chains on via a comparison theorem which is of independent interest.